Chapter 3.5, Problem 72E

Solution

To justify: The conclusion made about natural logarithmic regression in the three types of logarithmic re-expression.

Given information: The three types of logarithmic re-expression.

Explanation:

Let assume a set of data points $\left(x,y\right)$ , where $x>0$ for all data.

Let define a new set of data points $\left(u,v\right)$ , where the $y$ values are the same as in the first set, and $u=\ln x$ for each corresponded point.

Assume that the points $\left(u,v\right)$ are a good fit for a linear regression equation, that is, they are arranged in a straight line in a scatter plot, or close enough to a straight line for a linear regression to be a viable option.

Then find a linear regression equation in the form;

  $y=au+b$

Where $a$ and $b$ are regression coefficients such that $a\ne 0$ .

As set $u=\ln x$ , make that substitution in the regression equation, obtained

  $y=a\ln x+b$

Which will then serve as a logarithmic regression equation for the original data point $\left(x,y\right)$ .